Is 4x^2+y^2+16x-6y+9=0 a parabola, a circle, an ellipse, or a hyperbola?

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Is 4x^2+y^2+16x-6y+9=0 a parabola, a circle, an ellipse, or a hyperbola?

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Answer 1 · 2018-05-11T08:45:47

An ellipse with vertical major axis.

Explanation:

$4 x^{2} + y^{2} + 16 x - 6 y + 9 = 0$ $4x^2+y^2+16x-6y+9=0$

Convert to standard form by completing the square:

$4 x^{2} + 16 x + y^{2} - 6 y = - 9$ $4x^2+16x+y^2-6y=-9$

$4 (x^{2} + 4 x + 4) + (y^{2} - 6 y + 9) = - 9 + 16 + 9$ $4(x^2+4x+4)+(y^2-6y+9)=-9+16+9$

$4 {(x + 2)}^{2} + {(y - 3)}^{2} = 16$ $4(x+2)^2+(y-3)^2=16$

$\frac{{(x + 2)}^{2}}{4} + \frac{{(y - 3)}^{2}}{16} = 1$ $(x+2)^2/4+(y-3)^2/16=1$

Equation of an ellipse with vertical major axis in standard form:
$\frac{{(x - h)}^{2}}{a^{2}} + \frac{{(y - k)}^{2}}{b^{2}} = 1$ $(x-h)^2/a^2+(y-k)^2/b^2=1$
Where:
$(h, k)$ $(h, k)$ is the coordinate of the center
$2 b$ $2b$ is the major axis

So in this case:
Center is at $(- 2, 3)$ $(-2, 3)$
$a = 2, b = 4$ $a=2, b=4$
Major axis is along the y-axis and the length is: $8$ $8$

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Is 4x^2+y^2+16x-6y+9=0 a parabola, a circle, an ellipse, or a hyperbola?

1 Answer

Explanation:

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